# Properties

 Label 32.0.15272562598...5488.1 Degree $32$ Signature $[0, 16]$ Discriminant $2^{191}\cdot 17^{16}$ Root discriminant $258.22$ Ramified primes $2, 17$ Class number Not computed Class group Not computed Galois group $C_{32}$ (as 32T33)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![97322383751333736962, 0, 732780301186512843008, 0, 915975376483141053760, 0, 452599597791669697152, 0, 117428677158132789072, 0, 18420184652256123776, 0, 1896195478908718624, 0, 134829672062029120, 0, 6840623067852948, 0, 252479744142208, 0, 6839621551840, 0, 135851917760, 0, 1954391400, 0, 19809216, 0, 134096, 0, 544, 0, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 544*x^30 + 134096*x^28 + 19809216*x^26 + 1954391400*x^24 + 135851917760*x^22 + 6839621551840*x^20 + 252479744142208*x^18 + 6840623067852948*x^16 + 134829672062029120*x^14 + 1896195478908718624*x^12 + 18420184652256123776*x^10 + 117428677158132789072*x^8 + 452599597791669697152*x^6 + 915975376483141053760*x^4 + 732780301186512843008*x^2 + 97322383751333736962)

gp: K = bnfinit(x^32 + 544*x^30 + 134096*x^28 + 19809216*x^26 + 1954391400*x^24 + 135851917760*x^22 + 6839621551840*x^20 + 252479744142208*x^18 + 6840623067852948*x^16 + 134829672062029120*x^14 + 1896195478908718624*x^12 + 18420184652256123776*x^10 + 117428677158132789072*x^8 + 452599597791669697152*x^6 + 915975376483141053760*x^4 + 732780301186512843008*x^2 + 97322383751333736962, 1)

## Normalizeddefining polynomial

$$x^{32} + 544 x^{30} + 134096 x^{28} + 19809216 x^{26} + 1954391400 x^{24} + 135851917760 x^{22} + 6839621551840 x^{20} + 252479744142208 x^{18} + 6840623067852948 x^{16} + 134829672062029120 x^{14} + 1896195478908718624 x^{12} + 18420184652256123776 x^{10} + 117428677158132789072 x^{8} + 452599597791669697152 x^{6} + 915975376483141053760 x^{4} + 732780301186512843008 x^{2} + 97322383751333736962$$

magma: DefiningPolynomial(K);

sage: K.defining_polynomial()

gp: K.pol

## Invariants

 Degree: $32$ magma: Degree(K);  sage: K.degree()  gp: poldegree(K.pol) Signature: $[0, 16]$ magma: Signature(K);  sage: K.signature()  gp: K.sign Discriminant: $$152725625984366375633872344931707463035520335286488090842213114237373265215488=2^{191}\cdot 17^{16}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $258.22$ magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));  sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $2, 17$ magma: PrimeDivisors(Discriminant(Integers(K)));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ This field is Galois and abelian over $\Q$. Conductor: $$2176=2^{7}\cdot 17$$ Dirichlet character group: $\lbrace$$\chi_{2176}(1,·), \chi_{2176}(1155,·), \chi_{2176}(137,·), \chi_{2176}(1291,·), \chi_{2176}(273,·), \chi_{2176}(1427,·), \chi_{2176}(409,·), \chi_{2176}(1563,·), \chi_{2176}(545,·), \chi_{2176}(1699,·), \chi_{2176}(681,·), \chi_{2176}(1835,·), \chi_{2176}(817,·), \chi_{2176}(1971,·), \chi_{2176}(953,·), \chi_{2176}(2107,·), \chi_{2176}(1089,·), \chi_{2176}(67,·), \chi_{2176}(1225,·), \chi_{2176}(203,·), \chi_{2176}(1361,·), \chi_{2176}(339,·), \chi_{2176}(1497,·), \chi_{2176}(475,·), \chi_{2176}(1633,·), \chi_{2176}(611,·), \chi_{2176}(1769,·), \chi_{2176}(747,·), \chi_{2176}(1905,·), \chi_{2176}(883,·), \chi_{2176}(2041,·), \chi_{2176}(1019,·)$$\rbrace$ This is a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $\frac{1}{17} a^{2}$, $\frac{1}{17} a^{3}$, $\frac{1}{289} a^{4}$, $\frac{1}{289} a^{5}$, $\frac{1}{4913} a^{6}$, $\frac{1}{4913} a^{7}$, $\frac{1}{83521} a^{8}$, $\frac{1}{83521} a^{9}$, $\frac{1}{1419857} a^{10}$, $\frac{1}{1419857} a^{11}$, $\frac{1}{24137569} a^{12}$, $\frac{1}{24137569} a^{13}$, $\frac{1}{410338673} a^{14}$, $\frac{1}{410338673} a^{15}$, $\frac{1}{6975757441} a^{16}$, $\frac{1}{6975757441} a^{17}$, $\frac{1}{118587876497} a^{18}$, $\frac{1}{118587876497} a^{19}$, $\frac{1}{2015993900449} a^{20}$, $\frac{1}{2015993900449} a^{21}$, $\frac{1}{34271896307633} a^{22}$, $\frac{1}{34271896307633} a^{23}$, $\frac{1}{582622237229761} a^{24}$, $\frac{1}{582622237229761} a^{25}$, $\frac{1}{9904578032905937} a^{26}$, $\frac{1}{9904578032905937} a^{27}$, $\frac{1}{168377826559400929} a^{28}$, $\frac{1}{168377826559400929} a^{29}$, $\frac{1}{2862423051509815793} a^{30}$, $\frac{1}{2862423051509815793} a^{31}$

magma: IntegralBasis(K);

sage: K.integral_basis()

gp: K.zk

## Class group and class number

Not computed

magma: ClassGroup(K);

sage: K.class_group().invariants()

gp: K.clgp

## Unit group

magma: UK, f := UnitGroup(K);

sage: UK = K.unit_group()

 Rank: $15$ magma: UnitRank(K);  sage: UK.rank()  gp: K.fu Torsion generator: $$-1$$ (order $2$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);  sage: UK.torsion_generator()  gp: K.tu[2] Fundamental units: Not computed magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: Not computed magma: Regulator(K);  sage: K.regulator()  gp: K.reg

## Galois group

$C_{32}$ (as 32T33):

magma: GaloisGroup(K);

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

 A cyclic group of order 32 The 32 conjugacy class representatives for $C_{32}$ Character table for $C_{32}$ is not computed

## Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R $32$ $32$ $16^{2}$ $32$ $32$ R $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{4}$ $32$ $32$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

magma: idealfactors := Factorization(p*Integers(K)); // get the data

magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

gp: idealfactors = idealprimedec(K, p); \\ get the data

gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$17$17.8.4.2$x^{8} - 4913 x^{2} + 918731$$2$$4$$4$$C_8$$[\ ]_{2}^{4} 17.8.4.2x^{8} - 4913 x^{2} + 918731$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
17.8.4.2$x^{8} - 4913 x^{2} + 918731$$2$$4$$4$$C_8$$[\ ]_{2}^{4} 17.8.4.2x^{8} - 4913 x^{2} + 918731$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$